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Merged
mikecroucher merged 2 commits into from
Apr 9, 2020
Merged

Improvements to the inhom_iema notebook #15

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751f93c
added equations and info
caleb-hamilton Apr 2, 2020
20f4f2d
formatting fix
caleb-hamilton Apr 2, 2020
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32 changes: 29 additions & 3 deletions time_series_analysis/inhom_iema.ipynb
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Original file line number Diff line number Diff line change
Expand Up @@ -5,9 +5,29 @@
"metadata": {},
"source": [
"# Iterated Exponential Moving Average on Inhomogeneous Time Series\n",
"This notebook calculates an Iterative Exponential Moving Average on a simulated Inhomogeneous Timer Series for 3 Values of $\\tau$ at once.\n",
"This notebook calculates an Iterative Exponential Moving Average on a simulated Inhomogeneous Timer Series for three rates of decay ($\\tau$) at once.\n",
"\n",
"See the NAG Library for Python documentation for the further details on this function: https://www.nag.com/numeric/py/nagdoc_latest/naginterfaces.library.tsa.html#naginterfaces.library.tsa.inhom_iema"
"The `inhom_iema` routine is using a decay kernel of the following form:\n",
"\n",
"$$\n",
"\\frac{e^{-t_i/\\tau}}{\\tau}\n",
"$$\n",
"\n",
"This gives rise to the following iterative formula of the Exponential Moving Average (EMA) operator:\n",
"\n",
"$$\n",
"\\text{EMA}[\\tau;z](t_i)=\\mu\\text{EMA}[\\tau;z](t_{i-1})+(\\nu-\\mu)z_{i-1}+(1-\\nu)z_i\n",
"$$\n",
"\n",
"where $t$ is the inhomogenous (although ordered) points in time, $z$ is the set of observations, and:\n",
"\n",
"$$\n",
"\\mu = e^{\\alpha} \\;\\; \\text{and} \\;\\; \\alpha = \\frac{t_i - t_{i-1}}{\\tau}\n",
"$$\n",
"\n",
"The value of $\\nu$ depends on the method of interpolation chosen.\n",
"\n",
"See the NAG Library for Python documentation for the further details on this function: [NAG Library for Python docs](https://www.nag.com/numeric/py/nagdoc_latest/naginterfaces.library.tsa.html#naginterfaces.library.tsa.inhom_iema)"
]
},
{
Expand Down Expand Up @@ -69,7 +89,13 @@
"\n",
"* `tau` - the decay parameter. this script will calculate the IEMA for 3 $\\tau$ values at once\n",
"* `sinit` - the first two elements represent an initial point in time and a corresponding observation, while the additional elements after represent initial values for the EMA for the total number of iterations you require (e.g. if sinit has 3 elements, there will only be one iteration)\n",
"* `inter` - select the interpolation method for the first and subsequent iterations, respectively; see docs for more info"
"* `inter` - select the interpolation method for the first and subsequent iterations, respectively; see docs for more info\n",
" \n",
" 1) Previous point: $\\nu = 1$\n",
" \n",
" 2) Linear: $\\nu = (1-\\mu) / \\alpha$\n",
" \n",
" 3) Next point: $\\nu = \\mu$"
]
},
{
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